\xiti\mylabel{xiti-8}

\begin{xiaotis}

\xiaoti{展开下列行列式，并化简：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.2}
    \begin{tabular}[t]{*{2}{@{}p{16em}}}
        \xiaoxiaoti{
            $\begin{vmatrix}
                x-1 & x^3 \\
                1   & x^2+x+1
            \end{vmatrix}$；}
        & \xiaoxiaoti{
            $\begin{vmatrix}
                \sin x - \sin y  & \cos x + \cos y \\
                \cos x - \cos y  & \sin x + \sin y
            \end{vmatrix}$；} \\
        \xiaoxiaoti{
            $\begin{vmatrix}
                1 - \sqrt{2}  & 2 - \sqrt{3} \\
                2 + \sqrt{3}  & 1 + \sqrt{2}
            \end{vmatrix}$；}
        & \xiaoxiaoti{
            $\begin{vmatrix}
                \log_a b  & 1 \\
                2         & \log_b a
            \end{vmatrix}$；} \\
        \xiaoxiaoti{
            $\begin{vmatrix}
                a - b  & a^2 - ab + b^2 \\
                a + b  & a^2 + ab + b^2
            \end{vmatrix}$；}
        & \xiaoxiaoti{
            $\begin{vmatrix}
                e^{x+y} & e^x - 1 \\
                e^x + 1 & e^{x-y}
            \end{vmatrix}$。} \\
    \end{tabular}

\end{xiaoxiaotis}


\xiaoti{求证：}\mylabel{xiti-8-2}
\begin{xiaoxiaotis}

    \xiaoxiaoti{
        $\begin{vmatrix}
            a_1 & b_1 \\
            a_2 & b_2
        \end{vmatrix} = \begin{vmatrix}
            a_1 & a_2 \\
            b_1 & b_2
        \end{vmatrix}$；
    }

    \xiaoxiaoti{
        $\begin{vmatrix}
            b_1 & a_1 \\
            b_2 & a_2
        \end{vmatrix} = - \begin{vmatrix}
            a_1 & b_1 \\
            a_2 & b_2
        \end{vmatrix}$；
    }

    \xiaoxiaoti{
        $\begin{vmatrix}
            ka_1 & b_1 \\
            ka_2 & b_2
        \end{vmatrix} = k \begin{vmatrix}
            a_1 & b_1 \\
            a_2 & b_2
        \end{vmatrix}$；
    }

    \xiaoxiaoti{
        $\begin{vmatrix}
            la_1 & ka_1 \\
            la_2 & ka_2
        \end{vmatrix} = 0$；
    }

    \xiaoxiaoti{
        $\begin{vmatrix}
            a_1 + a_1'  & b_1 + b_1' \\
            a_2         & b_2
        \end{vmatrix} = \begin{vmatrix}
            a_1 & b_1 \\
            a_2 & b_2
        \end{vmatrix} + \begin{vmatrix}
            a_1' & b_1' \\
            a_2  & b_2
        \end{vmatrix}$；
    }

    \xiaoxiaoti{
        $\begin{vmatrix}
            a_1 + ka_2  & b_1 + kb_2 \\
            a_2         & b_2
        \end{vmatrix} = \begin{vmatrix}
            a_1 & b_1 \\
            a_2 & b_2
        \end{vmatrix}$。
    }

\end{xiaoxiaotis}


\xiaoti{利用行列式解下列方程组：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.2}
    \begin{tabular}[t]{*{2}{@{}p{16em}}}
        \xiaoxiaoti{
            $\begin{cases}
                13x - 7y - 10 = 0, \\
                19x + 15y - 2 = 0;
            \end{cases}$
        } & \xiaoxiaoti{
            $\begin{cases}
                \dfrac{7}{s} + \dfrac{9}{t} = 3, \\[1.5em]
                \dfrac{17}{s} + \dfrac{7}{t} = 5 \text{。}
            \end{cases}$
        }
    \end{tabular}

\end{xiaoxiaotis}

\xiaoti{利用行列式解下列关于 $x$，$y$ 的方程组：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.2}
    \begin{tabular}[t]{*{2}{@{}p{16em}}}
        \xiaoxiaoti{
            $\begin{cases}
                mx + y = 2m + 1, \\
                x - my = 2 - m;
            \end{cases}$
        } & \xiaoxiaoti{
            $\begin{cases}
                x \cos A - y \sin A = \cos B, \\
                x \sin A + y \cos A = \sin B;
            \end{cases}$
        }
    \end{tabular}

    \xiaoxiaoti{
        $\begin{cases}
            x \cos A + y \sin A = \sin A, \\
            x \sin A + y \cos A = -\cos A;
        \end{cases} \quad \left( A \neq \dfrac{2k + 1}{4}\pi ,\; k \in Z \right)$。
    }

\end{xiaoxiaotis}


\xiaoti{不解方程组，判定下列方程组有唯一解，无解，还是有无穷多解：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.2}
    \begin{tabular}[t]{*{2}{@{}p{16em}}}
        \xiaoxiaoti{
            $\begin{cases}
                2x + 3y = 7, \\
                5x - 2y = 1;
            \end{cases}$
        } & \xiaoxiaoti{
            $\begin{cases}
                6x + 9y = 7, \\
                4x + 6y = 2;
            \end{cases}$
        } \\[2em]
        \xiaoxiaoti{
            $\begin{cases}
                4x - 3y = 5, \\
                8x + 6y = 22;
            \end{cases}$
        } & \xiaoxiaoti{
            $\begin{cases}
                5x - 15y = 10, \\
                3x - 9y = 6 \text{。}
            \end{cases}$
        }
    \end{tabular}

\end{xiaoxiaotis}


\xiaoti{判断 $m$ 取什么值时，下列关于 $x$，$y$ 的方程组有唯一解：}
\begin{xiaoxiaotis}

    \xiaoxiaoti{
        $\begin{cases}
            (m^2 - 1)x - (m + 1)y = m + 1, \\
            m^2 x - (m + 1)y = m - 1;
        \end{cases}$
    }

    \xiaoxiaoti{
        $\begin{cases}
            x - (m^2 - 5)y = -1, \\
            (m + 1)x - (m + 1)^2 y = 1 \text{。}
        \end{cases}$
    }

\end{xiaoxiaotis}


\xiaoti{解下列关于 $x$，$y$ 的方程组，并进行讨论：}
\begin{xiaoxiaotis}

    \xiaoxiaoti{
        $\begin{cases}
            ax + (2a - 1)y = a^2 + 2a - 1, \\
            x + ay = 2a;
        \end{cases}$
    }

    \xiaoxiaoti{
        $\begin{cases}
            mx + y = -1, \\
            3mx - my = 2m + 3;
        \end{cases}$
    }

    \xiaoxiaoti{
        $\begin{cases}
            (a - 1)x - (a^2 - 1)y = 1, \\
            (a^2 - 1)x + (a - 1)y = 2 \text{。}
        \end{cases}$
    }

\end{xiaoxiaotis}


\end{xiaotis}

